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03.2.1. Fixed Effects Estimation of the Population-Averaged Slopes in a Panel Data Random Coefficient Model

Published online by Cambridge University Press:  01 April 2003

Jeffrey M. Wooldridge
Jeffrey M. Wooldridge
Affiliation:
Michigan State University, East Lansing
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Abstract

Fixed effects estimation of the population-averaged slopes in a panel data random coefficient model.

Information

Type
PROBLEMS AND SOLUTIONS
Information
Econometric Theory , Volume 19 , Issue 2 , April 2003 , pp. 411 - 412
Copyright
© 2003 Cambridge University Press

Consider a linear panel data model with individual-specific slopes and also an individual-specific intercept. For a random draw i from the cross section, the model is

where xit is 1 × K and bi is K × 1. You are interested in estimating β ≡ E(bi), the population-averaged slope coefficients. In what follows, make the strict exogeneity assumption

Let i denote the 1 × K vector of time averages and let

be the vector of time-demeaned variables. The T × K matrix

contains the

as its T rows. If bi = β then it is well known that the fixed effects, or within, estimator consistently estimates β, as N → ∞, holding T fixed, under assumption (2) and the rank condition

Essentially, condition (3) rules out time-constant regressors in addition to perfect collinearity. See, for example, Wooldridge (2002, Ch. 10). This problem asks you to study the properties of the fixed effects estimator in the more general model, (1).

a. Define di ≡ bi − β. Write equation (1) in terms of di and β.

b. Let

denote the time-demeaned quantities for the dependent variable and similarly for

. Write the time-demeaned equation in terms of

.

c. Show that under assumptions (2), (3), and

the fixed effects estimator is consistent for β. Explain why assumption (4) allows bi to be correlated with the time average, i, and why allowing such correlation may be important in practice.

d. Can we relax (4) to the assumption that bi is uncorrelated with each

and still maintain consistency? Explain.

References

REFERENCE

Wooldridge, J.M. (2002) Econometric Analysis of Cross Section and Panel Data. Cambridge, MA: MIT Press.